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LP—Graphical Solution Method MSci331—Week 2-3 1

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Convex Set and Extreme Points 2

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LP: Example (Papa Louis ) Papa Louis manufacturers wooden tables and chairs for small kids. Each "table" built: Sells for $27 and uses $10 worth of raw materials, increases Papa Louis’s variable labor/overhead costs by $14. Requires 2 hours of finishing labor and 1 hour of carpentry labor. Each "chair" built: Sells for $21 and uses $9 worth of raw materials, increase Papa Louis’s variable labor/overhead costs by $10. Requires 1 hours of finishing labor AND 1 hour of carpentry labor. Each week Papa Louis can obtain only 100 finishing hours and only 80 carpentry hours. Also demand for the chairs is unlimited. However, at most 40 tables are bought each week. Papa Louis wants to maximize weekly profit (revenues - expenses). 3

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LP: Example (Papa Louis ) 4

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8 Example: 2

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Example 2: Multiple Optimal Solutions Consider the LP model 9

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Example 2: Multiple Optimal Solutions 10

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Infeasible Solution 11

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Unbounded Solution 12

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LP Model to Standard Form Convert LP format to a standard form 13

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Insight from the Geometric Procedure For a constraint to be reasonable, all terms in the constraints must have the same units. 14

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403530252015105 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 Insights from the Geometric Procedure x1x1x1x1 11 22 33 99 55 x 2 x 2 66 Constraint 1 Constraint 2 Constraint 3 88 1010 Z 77 11 (x 1 =0,x 2 =0,Z=0) 22 (x 1 =0,x 2 =5,Z=15) 88 (x 1 =5,x 2 =10,Z=40) 1010 (x 1 =20,x 2 =5,Z=55) 77 (x 1 =20,x 2 =0,Z=40) 15

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Basic and Nonbasic Variables 16

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Basic and Nonbasic Variables 17

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Division ÷ 1 1 ÷ 1 = 1 2 ÷ 1 = 2 3 ÷ 1 = 3 4 ÷ 1 = 4 5 ÷ 1 = 5 6 ÷ 1 = 6 7 ÷ 1 = 7 8 ÷ 1 = 8 9 ÷ 1 = 9 10 ÷ 1 = 10 11 ÷ 1 = 11 12 ÷ 1 = 12 ÷ 2 2 ÷ 2 =

Division ÷ 1 1 ÷ 1 = 1 2 ÷ 1 = 2 3 ÷ 1 = 3 4 ÷ 1 = 4 5 ÷ 1 = 5 6 ÷ 1 = 6 7 ÷ 1 = 7 8 ÷ 1 = 8 9 ÷ 1 = 9 10 ÷ 1 = 10 11 ÷ 1 = 11 12 ÷ 1 = 12 ÷ 2 2 ÷ 2 =

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